difference of sets?

Question

what is the difference between subsets and proper subsets? other than the difference in sign?

Answer 1

The set {1, 2} is a proper subset of {1, 2, 3}.

The set {1, 2, 3} is NOT a proper subset of {1, 2, 3} but it is a subset of {1, 2, 3}.

A proper subset will not include every element in the superset.
So, an empty set will be a proper subset of any set with at least one element.

Get it?

Answer 2

Definition of a subset:
A is a subset of B if for every x in A, x is in B.

Definition of a proper subset:
A is a proper subset of B if for every x in A, x is in B and there exists some y in B such that y isn't in A.

Answer 3

A proper subset of a set S is one that is strictly contained inside S. This is different from the regular definition of 'subset' which allows S to be a subet of itself. A proper subset MUST exclude at least 1 element of S.

So consider the set, S={0,1,2}
Then the subsets of S are:
{0}, {1}, {2}, {0,1}, {0,2}, {1,2}, {0,1,2}

However, the proper subsets are:
{0}, {1}, {2}, {0,1}, {0,2}, {1,2}

Notice that {0,1,2} is NOT a proper subset. It does not exclude at least 1 element of S.